A state-space mark-recapture model for simultaneously estimating collection efficiency and abundance
John Best
Quantitative Synthesis and Reporting
Thomas Buehrens
Fish Science Region 4
Cowlitz River
Lake Scanewa
Estimating Fish Collection Efficiency
Current method
- Only tags when 100 juveniles of a species are available
- Simplistic method of moments estimator
- Arithmetic mean used to estimate season-long collection efficiency
Estimating Fish Collection Efficiency
Current method
- Only tags when 100 juveniles of a species are available
- Simplistic method of moments estimator
- Arithmetic mean used to estimate season-long collection efficiency
Problems
- Limited temporal coverage
- Coarse temporal resolution
- Doesn’t weight by arrivals
Estimating Fish Collection Efficiency
Components
- Number of fish collected
- Number of fish arriving
Estimating Fish Collection Efficiency
Components
- Number of fish collected
- Number of fish arriving
Estimation goals
- Daily probability of capture
- Daily number of fish arriving
What about BTSPAS1?
- Typically stratified by week
- Includes an explicit movement/delay component
- Model for probability of capture includes (independent) random effects
- Model for arrival rate depends on a temporal smoother
- Fit using JAGS2
Probability of Capture
Travel Time
- \(N_r\) fish are released on day \(r\)
- Travel time is modeled as a function \(F_{\theta}\)
- Proportion released on day \(r\) arriving on day \(t\) is
\[\theta_{rt} = \begin{cases}
0 & t \le r\\
F_{\theta}(t - r) - F_{\theta}(t - r - 1) & r < t, t - r \le T\\
0 & i - t > T
\end{cases}\]
- Assume no arrivals after day \(T\)
Probability of Capture
Proportion arriving each day
Probability of Capture
Probability of collection
- Given arrival, how likely is an individual to be collected?
- Modeled as \(p_t\), probability of collection on day \(t\)
- Flexible specification, allows covariates \(\boldsymbol{x}_t\)
- May use
mgcv smoothers, including for time
\[p_{t} = \operatorname{logit}^{-1}(\phi_t) = f_{\phi}(\boldsymbol{x}_{t})\]
Probability of Capture
Observation Likelihood
For
- \(n_{rt}\) marked fish recaptured from release \(r\) on day \(t\) after release
- \(N_{r}\) total number of fish released on day \(r\)
- \(\theta_{rt}\) proportion of fish arriving on day \(t\) from release \(r\)
- \(p_t\) probability of capture on day \(t\)
\[[n_{r1},\ldots,n_{rT}, N_{r} - \sum_{t} n_{rt}] \sim\] \[\operatorname{Multinomial}(\theta_{r1} p_{r+1}, \ldots, \theta_{rT} p_{r+T}, 1 - \sum_{t} \theta_{rt} p_{t}\]
Arrival Process
- Log-arrival rate \(\lambda_{t}\) is modeled as a function of covariates \(\boldsymbol{z}_t\),
\[\log(\lambda_{t}) = f_{\lambda}(\boldsymbol{z}_{t}),\]
- \(U_{t}\) unmarked fish arrive on day \(t\) at rate \(\lambda_{t}\), so that
\[U_{t} \sim \operatorname{Poisson}(\lambda_{t})\]
- \(u_{t}\) individuals are collected with probability \(p_{t}\), so
\[u_t \sim \operatorname{Poisson}(\lambda_{t} p_t)\]
Collection efficiency
- Simulate values of \(U_t\) from a truncated Poisson distribution
\[U_t^{*} \sim \operatorname{Poisson}(\lambda_t) \quad \text{s.t.}\quad U_t \ge u_t\]
- Then estimated Fish Collection Efficiency is
\[\widehat{\textrm{FCE}} = \frac{\sum_t u_t}{\sum_t U_t^*}\]
Priors
- Covariates are scaled to have mean zero, standard deviation one
- Each coefficient has a \(Normal(0, \sigma_{i}^{2})\) prior
- \(\sigma_{i}\) is not currently estimated, including for smooths (where it acts as the smoothness penalty)
Model fitting
- Model is fit using the No-U-Turn Sampler in Stan
Cowlitz River Steelhead
Data
- 2020 to 2022
- Six releases each year
- 100 individuals per release1
- No recaptures after 33 days
Cowlitz River Steelhead
Covariates
- Year intercept
- Two-way smooth with temperature and discharge
- Gaussian process smooth over the trapping season
Results
Partial effect of temperature and discharge
Results
Partial effect of temperature and discharge
Results
Estimated probability of capture
Results
Arrival rate estimate
Results
Run size estimate
Results
Fish Collection Efficiency
Can we do better?
- Assume temperature and discharge effects are shared among species
- Include separate intercepts and temporal smooths for Chinook and coho
- Adds 4 Chinook releases in 2020, 6 in 2021, and 7 in 2022
- Adds 11 coho release in 2020, 10 in 2021, and 9 in 2022
Results
Partial effect of temperature and discharge
Results
Estimated probability of capture
Results
Arrival rate estimate
Results
Run size estimate
Results
Fish Collection Efficiency
Future work
Other applications
- Other dam systems
- Smolt trap data
Improvements
- Allow travel time to vary over each season
- Account for more variation in arrival process
- Estimate smoothness penalty parameters
- Allow for batch-marked fish
- Allow for multiple recapture opportunities
- Create an R package for general use